Adobe After Effects is a compositing, compositional and post-production video tool. If you think about green-screening, those cool titles/credits, overlays with people's names, infographics... it's all that stuff, and a lot more. Like motion photoshop, in full hd.

After Effects is a compositing, compositional and post-production video tool. If that doesn't mean anything to you, then think about green-screening, those cool titles/credits, overlays with people's names, infographics... it's all that stuff, and a lot more. Like motion photoshop, in full hd.
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danpJul 21 '11 at 0:32

vote up danp comment please, because its the true and very usefull. I work with AE (After Effects) and wanna see this question growing up. JFR should copy this into his question too, imho. @danp
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H_7Mar 3 '12 at 15:26

+1 for Ramen. That one is new to me. I would also throw Blender into the mix. While it's better known as a 3d animation tool it also includes a node based compositor and NLE.
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3dinfluenceSep 3 '10 at 0:21

I've been an After Effects user for 12+ years and I have not found anything really comparable on linux platforms. Cinelerra is probably the closest and looks promising, but I have little experience with it (basically since my clients are dedicated to AE). The other apps listed by others above are primarily video editors; and as I'm sure you're aware, that is not After Effects primary niche.

Cinepaint, developed for the motion picture industry says that they may eventually add features in the direction of AE.

Although this is a rather old question, for the benefit of new users / searchers, I think blender needs a more prominent place in the answers than as in a comment.
With the finalization of project mango, called Tears of Steel, blenders abilities in regard to post production are stronger than ever -- the node based compositor is very powerful indeed. You need to get used to blender, but it's worth it.

Update: Blender 2.69 was just released a couple of days ago, with many improvements in camera tracking, motion tracking, ...